Open AccessTHE CAUCHY-DIRICHLET PROBLEM FOR A SYSTEM OF FIRST-ORDER EQUATIONS
Author(s) -
Zh. A. Tokibetov,
N. E. Bashar,
А.К. Pirmanova
Publication year - 2020
Publication title -
habaršy. fizika-matematika seriâsy
Language(s) - English
Resource type - Journals
ISSN - 1728-7901
DOI - 10.51889/2020-4.1728-7901.10
Subject(s) - elliptic partial differential equation , mathematics , hyperbolic partial differential equation , cauchy boundary condition , mathematical analysis , laplace's equation , boundary value problem , cauchy problem , dirichlet problem , partial differential equation , initial value problem , d'alembert's formula , first order partial differential equation , elliptic boundary value problem , green's function for the three variable laplace equation , dirichlet boundary condition , differential equation , free boundary problem , ordinary differential equation , differential algebraic equation
For a single second-order elliptic partial differential equation with sufficiently smooth coefficients, all classical boundary value problems that are correct for the Laplace equations are Fredholm. The formulation of classical boundary value problems for the laplace equation is dictated by physical applications. The simplest of the boundary value problems for the Laplace equation is the Dirichlet problem, which is reduced to the problem of the field of charges distributed on a certain surface. The Dirichlet problem for partial differential equations in space is usually called the Cauchy-Dirichlet problem. This work dedicated to systems of first-order partial differential equations of elliptic and hyperbolic types consisting of four equations with three unknown variables. An explicit solution of the CauchyDirichlet problem is constructed using the method of an exponential – differential operator. Giving a very simple example of the co-solution of the Cauchy problem for a second-order differential equation and the Cauchy problem for systems of first-order hyperbolic differential equations.